Help on the expectation value of two added operators

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The discussion centers on the expectation value of two linear operators A and B in quantum mechanics, specifically whether the equation < ψ | (A+B) | ψ > = < ψ | A | ψ > + < ψ | B | ψ > holds true for any state ket | ψ >. The original poster expresses uncertainty about this property and seeks clarification on the definition of linear operators. They attempt to prove the equation by showing that (A+B) | ψ > = A | ψ > + B | ψ > but struggle to find a definitive source. A helpful suggestion is made to refer to P.A.M. Dirac's "Principles of Quantum Mechanics," which addresses linear operators in detail. The conversation highlights the importance of understanding linear operators in quantum mechanics.
grzegorz19
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Hi everyone,

I was just working on some problems regarding the mathematical formalism of QM, and while trying to finish a proof, I realized that I am not sure if the following fact is always true:

Suppose that we have two linear operators A and B acting over some vector space. Consider a state ket | \psi >

I am wondering if
< \psi | (A+B) | \psi > = < \psi | A | \psi > + < \psi | B | \psi >
is always true?

I am thinking that it IS true.

My attempt at the problem, is of course to try and show that
(A+B) | \psi > = A | \psi > + B | \psi >
But I am having trouble finding a definition which will confirm this to always be true.

I feel like I am completely overlooking something. Does anyone have a helpful hint for me? ANy literature to point me to? My linear algebra books are failing me on this one, at first glance.
 
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Definition of a linear operator?
 
grzegorz19 said:
I feel like I am completely overlooking something. Does anyone have a helpful hint for me? ANy literature to point me to?
Checkout Principles of Quantum Mechanics (P.A.M. Dirac) chapter II, Dynamical Variables and Observables, section 7, Linear Operators.
 
THANK YOU! I don't know why this was so hard to find, but this is exactly the sort of thing I was looking for!
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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